Has Zhang's work on bounded gaps between primes been extended to the following theorem?

For any arithmetic progression $an+b,\gcd(a,b)=1$, there is a constant $H$ (depending only on $a$) such that there are infinitely many prime pairs $p_1<p_2$ in this progression with $p_2-p_1<H$.

I have tried to look for this result online, the closest hit I got is this paper, which proves similar sounding statement.

How about extending this question to sets of primes of positive relative density (this extension and question as the whole inspired by answer to this question) and quantitative bounds?

Thanks in advance.